Spectral Properties Of A Class Of Sturm - Liouville Equations With Transition

Recently, more and more researchers are interested in the discontinuous Sturm-Liouville problem for its wide application in physics area and engineer. More and more people pay attention and research such problems. As we all know, the solutions and quasi-derivatives are absolutely continuous on the compact subset of the problem interval. However,many problems don’t satisfy this condition, such as heat and mass transfer(see [5]), vibrating string problems when the string loaded additionally with point masses, the heat transfer problems of the laminated plate of membrane(that is, the plate which is formed by overlap of materials with different characteristics) and diffraction problems. Particularly, more and more people have paid close attention to Sturm-Liouville problem of the boundary condition depending on eigenparameter, the asymptotic of eigenvalues and eigenfunctions, shaking theory, etc, are studied. The various physics applications of this kind of problem are found in many literature([7-9, 11-16, 19-23]). Inspired by [11][12][18][19], we given a special weight function, considered spectral properties for Sturm-Liouville equations with transmission conditions and eigenparameter-dependent boundary conditions.By using the classical analysis techniques and spectral theory of linear operator, we de?ne a new self-adjoint operator associated with the problem in a new Hilbert space, such that the eigenvalues of such a problem is coincided with the operator. By searching the general solution of Sturm-Liouville equation and using boundary conditions and transmission conditions, obtain the fundamental solutions and the characteristic function of the problem, also get the asymptotic formulas for its eigenvalues and eigenfunctions, meanwhile, discuss the completeness of the eigenfunctions, and obtain its Green function and the resolvent operator.The thesis is divided into two chapters according to contents.Chapter 1 Preference, simply summarize the phylogeny of SturmLiouville theory and introduce the work of present paper.Chapter 2 we shall consider the discontinuous eigenvalue problem which consist of Sturm-Liouville equationτ u :=-u′′(x) + q(x)u(x) = λω(x)u(x)on x ∈ J = [a, ξ) ∪(ξ, b], with eigenparameter dependent boundary conditions at endpoints L1 u := λ(α′1u(a)- α′2u′(a))-(α1u(a)- α2u′(a)) = 0,L2 u := λ(β′1u(b)- β′2u′(b)) +(β1u(b)- β2u′(b)) = 0,and two transmission conditions at the point of discontinuity x = ξL3u := u(ξ + 0)- γ1u(ξ- 0)- δ1u′(ξ- 0) = 0,L4 u := u′(ξ + 0)- γ2u(ξ- 0)- δ2u′(ξ- 0) = 0,where q(x) is a given real-valued function continuous on [a, ξ), and(ξ, b](that is, continuous in J and has ?nite limits q(ξ ± 0) = lim x�'ξ±q(x)); λis a complex eigenvalue parameter; the coe?cients of the boundary and transmission conditions are real numbers; we de?ne ω(x) =1x2for x ∈ J.We assume further that ab > 0, ρ1= γ1γ2δ1δ2 > 0, ρ2= α′1α1α′2α2 > 0,ρ3= β′1β1β′2β2 > 0.